A transvestite, transgendered, or transexual mtf (male to female) person who has not yet shaved off his mustache. This gives the appearance of a woman with a mustache. This causes a conflict between the desired feminine appearance and the masculine mustache.
Stache clashes are often committed by males who are not yet ready to fully feminize themselves.

Fred: "Hey look, that woman has a mustache!"
Joe: "No, thats just a transvestite. Wow, what a stache clash..."
Fred:" Yeah, why dress up like a girl, but still keep the mustache?"